The volume of a semiregular, symmetrical shape like a pentagonal prism can be found using basic mathematical concepts and techniques. As with any prism, the volume can be calculated by finding the product of the area of the base multiplied by the height. The area of the pentagonal base is determined by a formula using the number of sides, the length of a side and a measurement known as the apothem, which is the perpendicular distance from any side to the center of the pentagon.

Make sure that the prism dimensions of base side length, apothem and height are all in the same units.

Calculate the perimeter of the prism base using the formula P = 5(s), where s is the length of any one side of the pentagon.

Calculate the area of the prism base using the formula A = (1/2)(P)(a), where P is the perimeter just calculated and a is the apothem of the base.

Multiply the base area just calculated by the height of the prism. The result is the volume of the prism.

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